Many people cite Latin when this debate is in full flow. They point out that the verb 'to be' has the subject and predicate in the nominative case. Thus 'This man is a scoundrel' would have 'man' and 'scoundrel' in the nominative case. Based on this argument, 'It is I' would be grammatically correct, and it is only usage that makes it wrong. Opponents would point out that we don't speak Latin, and therefore we do not have to follow the rules of Latin grammar. However, suppose that you have a longer sentence: It was I who was speaking OR It was me who was speaking. I would submit that the first sentence is correct. Another example: It is I who am driving the car. This sounds so odd that it would be better to rephrase it. I have heard newsreaders (when there are two of them) sign off like this to avoid the problem: So, it's goodbye from John and from me, rather than the simpler form: So, it's goodbye from John and me.
I have yet to hear an American on TV say 'me' in the sort of situation that harry supplies. They always say "between you and I", which is wrong in my opinion. But, given that there are more of them than there are of us, and that usage decides what is right and what is wrong, who knows if in 50 years' time we won't all be saying "between you and I"?